Forecasting secular variation using core flows
- Ciarán Beggan^{1}Email author and
- Kathy Whaler^{2}
Received: 30 November 2009
Accepted: 1 July 2010
Published: 31 December 2010
Abstract
Over the past ten years satellite measurements in combination with data from ground-based observatories have allowed very detailed models of the secular variation (SV) of the Earth’s magnetic field to be constructed. However, forecasting the change of the main field still remains a challenge, primarily because the core processes controlling SV are not sufficiently well understood. Hence, most forecasts do not appeal to any physical modelling constraints but use, for example, polynomial extrapolation from previous measurements. We attempt to apply a physical model to forecast the average SV during 2010–2015 by developing a core flow model. This steady flow model, derived from SV data during 2004.5 to 2009.5, generates a set of Gauss SV coefficients which are used to advect the large scale magnetic field forwards in time. Although this model has not been submitted as a candidate for IGRF-11, we present our SV prediction model and compare it to other candidate IGRF-11 SV models. In addition, we examine the use of the Ensemble Kalman filter to optimally assimilate field models derived from (1) forecast methods and (2) noisy data measurements. Such a scenario might conceivably arise if high quality satellite data with global coverage are not available for a significant period of time. We show that the overall misfit of the assimilated model to the actual field can be lower than the individual misfits of the input models, provided the uncertainties of each model are reasonably well known.
Key words
1. Introduction
Forecasting of magnetic field change has been attempted in many forms over the past 300 years since Halley’s observation of westward motion of the agonic line in the Atlantic hemisphere (Halley, 1692). Currently, a widely used forecast of the average secular variation (SV) is produced for the International Geomagnetic Reference Field (IGRF) model every five years. The IGRF models are an agreed set of field coefficients representing snapshots of the magnetic field at defined times and a set of coefficients forecasting the average SV for a period of five years into the future. The tenth generation of the International Geomagnetic Reference Field model (IGRF-10) covered the period from 2005.0 to 2010.0 (Macmillan and Maus, 2005). For the IGRF-10 SV model, the methods to estimate the average SV over its five year lifetime used a combination of polynomial extrapolation of satellite data and linear prediction filters applied to observatory data. However, these approaches do not invoke any particular physical arguments to support the assumption that the field coefficients will continue to change linearly (which, of course, they do not). Lowes (2000) succinctly outlines the issues faced in forecasting of SV, noting that an understanding of the fluid processes within the core might be useful in producing more accurate predictions. Since 2005, a number of forecasting techniques have been developed to incorporate physical approximations. For example, Sun et al. (2007) and Fournier et al. (2007) both outline frameworks in which observed magnetic field data can be assimilated into physical magnetohydrodynamic models. More recently, Kuang et al. (2009) have assimilated historical field data into numerical dynamo models to investigate if improvements can be made to forecast field models.
We present a forecast based upon a steady core flow model generated from satellite magnetic data measured over the period 2004.5–2009.5. The SV coefficients derived from a flow model are used to advect the main field Gauss coefficients forwards in time. In Section 2 we examine how well the IGRF-10 SV model forecast has performed while in Section 3 we compare the forecast from a steady flow model over a five year period starting in 2004.5 to the IGRF-10 estimate for 2009.5.
In Section 4, we investigate a method known as the Ensemble Kalman Filter to assimilate field models derived from magnetic measurements (from observatories or satellite) and field models computed from a forecast. Ensemble techniques to examine variability and error in geomagnetic studies have previously been applied to core flow modelling by Gillet et al. (2009). We outline our implementation scheme and use error estimates from comparisons to past IGRF and Definitive Geomagnetic Reference Field (DGRF) models to determine reasonable variances for the filter and to illustrate the potential benefits for forecasting in the absense of high-quality satellite magnetic data.
2. Forecasting Ability
The IGRF-10 model coefficients for 2005.0 were generated by forward extrapolation of data measured up to mid-2004 and are a combination of three candidate field models (Maus et al., 2005). Hence a revision is required to produce a retrospective DGRF model for 2005.0. The RMS difference between the IGRF-10 model and the DGRF model for 2005 is approximately 13 nT. The misfit of the IGRF-10 model coefficients to the best available estimate of the internal magnetic field is thus slightly larger than the 5 nT suggested in Maus et al. (2005).
It can be concluded that adequate techniques to accurately estimate the global SV over five years have not yet been developed. This is due, primarily, to the unknown changes of fluid flow within the core (e.g. Holme, 2007). In the next section, we attempt to improve upon the SV forecast by use of a steady core flow model to advect the magnetic field model forward in time.
3. Flow Modelling
We investigate whether using a core flow model to predict SV can reduce the RMS difference between the actual and predicted field at the end of five years. A study by Maus et al. (2008) examined how well hindcasting of the magnetic field using core flow models reproduced the observed field over 13 years. They employed the ‘frozen flux’ approximation in which diffusion of the magnetic field on large scales is assumed to be negligible on short timescales (Roberts and Scott, 1965). Maus et al. (2008) compared the prediction of the field back in time from the SV generated by a number of core flow modelling assumptions including steady, toroidal only and accelerated flows. They concluded that a steady flow produces the best average fit over a ten-year period. Following on from this result, Beggan and Whaler (2009) investigated how well a steady flow generated from satellite data could predict secular variation. Using the same techniques, here we test how well a steady core flow model developed from CHAMP satellite data from 2001.4–2004.5 can predict the change in the magnetic main field between 2004.5 and 2009.5. These time periods are covered by current satellite field models and allow consistent comparisons to be made.
For this study, we prepared a series of 27 monthly SV data sets, over the period 2001.4–2004.5, generated from CHAMP satellite data using the ‘Virtual Observatory’ (VO) method of Mandea and Olsen (2006). The SV data were inverted for toroidal and poloidal flow using the linear relationship between SV and flow spherical harmonic coefficients. The relation is through the Gaunt/Elsasser matrix (H) whose elements depend on the main field coefficients which change with time (Whaler, 1986). The main field, SV and flow coefficients are truncated at degree and order n_{max} = 14, and thus we have assumed that only large scale flows are responsible for the large scale SV. We assumed a steady flow model, with tangential geostrophy (Hills, 1979; Le Mouël, 1984) imposed as a weak penalty norm constraint. The Gauss coefficients (in the vector g) from the POMME3 main field model (Maus et al., 2006) were used in the flow inversion.
After five years, the RMS difference between the forecast model and the satellite field model CHAOS-2 (at 2009.5) was 85 nT, or approximately 17 nT/yr. As a comparison, the IGRF-10 SV coefficients were used to compute the field in 2009.5 using the CHAOS-2 coefficients for 2004.5 as the starting point. The misfit of the IGRF-10 SV model at 2009.5 to the CHAOS-2 model was 102 nT.
We verified that the flow model generated directly from the SV derived from the VO method is insensitive to the main field model used in the inversion by comparing the prediction of a steady flow derived from the CHAOS main field model (Olsen et al., 2006) instead of POMME3. The resulting flow model was only slightly different from that derived from the POMME3 model, and the RMS difference between the SV forecast model using the CHAOS-derived steady flow after five years was also approximately 85 nT. Thus, SV estimates from steady core flows give a lower RMS misfit over five years and suggest a forecasting method employing core flow modelling may be beneficial.
Predicted average SV coefficients from the steady flow model, up to degree and order 8.
n | m | ||
---|---|---|---|
1 | 0 | 12.5 | 0 |
1 | 1 | 16.9 | −28.2 |
2 | 0 | −11.4 | 0 |
2 | 1 | −5.4 | −21.3 |
2 | 2 | 2.0 | −12.1 |
3 | 0 | 0.9 | 0 |
3 | 1 | −3.2 | 7.9 |
3 | 2 | −3.4 | −3.4 |
3 | 3 | −7.5 | −2.2 |
4 | 0 | −1.6 | 0 |
4 | 1 | 1.8 | 0.4 |
4 | 2 | −8.2 | 3.6 |
4 | 3 | 5.0 | 3.0 |
4 | 4 | −2.5 | −0.9 |
5 | 0 | −0.6 | 0 |
5 | 1 | 0.5 | −0.1 |
5 | 2 | −1.7 | 1.7 |
5 | 3 | −0.9 | 1.1 |
5 | 4 | 1.4 | 3.7 |
5 | 5 | 0.8 | −0.8 |
6 | 0 | 0.2 | 0 |
6 | 1 | −0.3 | 0.3 |
6 | 2 | −0.3 | −1.6 |
6 | 3 | 2.0 | −0.2 |
6 | 4 | −1.5 | −0.2 |
6 | 5 | −0.3 | 0.6 |
6 | 6 | 2.0 | 0.4 |
7 | 0 | 0.2 | 0 |
7 | 1 | −0.1 | 0.6 |
7 | 2 | −0.7 | 0.2 |
7 | 3 | 1.1 | −0.2 |
7 | 4 | 0.1 | −0.2 |
7 | 5 | 0.1 | −0.7 |
7 | 6 | −0.8 | 0.2 |
7 | 7 | 0.8 | 0.2 |
8 | 0 | 0.3 | 0 |
8 | 1 | 0.0 | 0.0 |
8 | 2 | −0.3 | 0.3 |
8 | 3 | 0.3 | 0.0 |
8 | 4 | −0.3 | 0.3 |
8 | 5 | 0.3 | −0.1 |
8 | 6 | 0.3 | −0.2 |
8 | 7 | −0.6 | 0.6 |
8 | 8 | 0.1 | 0.3 |
4. Data Assimilation
We wish to investigate whether it is possible to combine a relatively good forecast field model with a lower quality ‘measured’ field model to improve the estimate of the actual field. For example, if the existing set of satellites fail before Swarm is fully operational, there could be a gap in which our present high-quality global coverage is diminished and we may become mostly reliant upon ground-based observatories and repeat stations to produce global magnetic field models. Due to the uneven geographic distribution of ground-based observations these models have much higher uncertainties than models employing satellite data. This section looks at a potential method for mitigating the impact of such an event by employing an optimal data assimilation algorithm to make best use of all available information. We investigate whether a sufficiently accurate forecast can be obtained using an initial high-resolution satellite field model, combined with a flow model for advection of the field and intermittent updates from a simulated lower quality ground-based field model.
Beggan and Whaler (2009) established a method for applying the Ensemble Kalman Filter (EnKF) technique of Evensen (1994) to combine SV forecasts from a steady flow model with intermittent updates from a relatively ‘noisy’ field model. As they noted, the results are highly dependent on the assumptions about the errors of the flow and field models. Indeed, if the field model is of high quality, then the EnKF technique is redundant. However, if the field model is relatively poor, then a combination of a forecast field model using a steady core flow model and a field model computed from ground-based data can reduce the overall misfit to the ‘true’ field.
4.1 Ensemble Kalman filtering
In our application of this method, we set to be a vector of Gauss coefficients produced from a forecast as defined in Eq. (2). If a field model, derived from observed data, becomes available it can be represented as z_{ t }, another vector of Gauss coefficients. Combining the measurement and the forecast depends on how the errors for each vector are defined, which we will discuss in Section 4.2.
Examination of Eqs. (3) and (4) reveals some important aspects of the filter. If the measurement z_{ t } and model prediction are equal, then the update has no effect. If the elements of the measurement covariance error matrix (Q) are small, then is more heavily weighted in the update. Equally, if the model covariance error matrix elements (P_{f}) are small then the forecast state is more important. The balance between these matrices (i.e. the estimated uncertainties of the model and measurement) controls the assimilation step.
The Kalman Filter was designed as a linear filter. In a non-linear regime, it can become unwieldy and unreliable. To overcome such limitations, Evensen (1994) suggested an algorithm for data assimilation employing Monte Carlo methods as an alternative. The state of a process at any particular time is represented as a vector in r-dimensional space, where r is the number of parameters in the system. In an Ensemble Kalman Filter the uncertainty of our knowledge of the process is represented by perturbing the inputs to the model forecast equations randomly by a known variance (with zero-mean) to produce an ‘ensemble’ of states—conceptually imagined as a ‘cloud’ of points in r-dimensional space. The evolution of the states though time is controlled by propagating the entire ensemble forward using model equations of the system behaviour. In essence, this is nominally equivalent to running many instances of the Kalman Filter in parallel.
When an ‘observation’ is available, it can be optimally assimilated into the ensemble by applying the standard Kalman Filter equations. With a sufficiently large ensemble (determined through experimentation), the mean state should represent the maximum likelihood value for the process at the time. The evolution of the ensemble can be explored by examining the ‘spread’ of the states about the mean.
There are three stages required to implement the EnKF for this problem: (1) generation of the initial ensemble, (2) forecasting the change of the field by driving the field model with SV predicted by core flow models and (3) assimilation of measurements e.g. from a ‘true’ field model. In our implementation (full details of the algorithm are given in Beggan and Whaler (2009)), the state ensemble (x_{ k }) is composed of a matrix of Gauss coefficients up to degree and order 12. The number of ensemble states was set to 1000 after experimentation, though it was found that any more than 500 is adequate.
The ensemble is initiated by perturbing the Gauss coefficients of the CHAOS-2 field model in 2004.5. The initial perturbation to the field coefficients is based on an estimate of the standard deviation for each coefficient, discussed in detail in the next section. A matrix of normally distributed random numbers N(0, 1) is generated, which is then multiplied by the standard deviation of the flow coefficients to give a perturbed flow coefficient matrix. This perturbed flow coefficient matrix is pre-multiplied by the H matrix to produce a matrix of perturbed SV coefficients, correctly scaled to reflect the uncertainty in the flow models. The perturbed SV coefficient matrix is then added to the initial state vector to produce the initial ensemble matrix. Once this initial ensemble has been created, forecasting and assimilation can take place.
The forecast (prediction) of the field is driven forwards by the summation of (1) the field coefficients and (2) the monthly SV coefficients from the flow model. In addition, at each timestep, model noise is added to simulate the variance of the ensemble, forcing it to grow at each forecast iteration. These steps are repeated until a measurement becomes available for assimilation into the ensemble.
Over time, the forecast field will begin to diverge from the actual field. To improve the forecast, data can be input into the ensemble to update (correct) it. The data have associated errors which are used to generate a perturbed data ensemble. Data, for example a set of Gauss coefficients (z_{ k }) with a certain (estimated or known) error are available. A matrix of zero-mean Gaussian random numbers is generated and scaled with the data error. The data are then added to the matrix of scaled random numbers to produce a matrix of ‘perturbed data’. Using Eq. (3) this data perturbation matrix and the perturbed SV coefficients are optimally assimilated into the ensemble at this timestep. The covariance matrices (P and Q) can be estimated from the ensemble and measurement errors (Evensen, 1994). In this study we assimilate low quality (i.e. large assumed variance; see Section 4.3) Gauss coefficients of the CHAOS-2 field model into the ensemble via the z_{ k } term.
4.2 Uncertainty estimation
The determination of realistic uncertainties to apply to the model coefficients of the field and the SV from the flow is difficult. For a Kalman Filter to be optimal it is important that the uncertainty estimate of the forecast field generated by the flow model and the uncertainty of the ‘measured’ field model should be similar in magnitude. If one has a much lower uncertainty estimate than the other, then it tends to dominate the filter.
Prior to the launch of Ørsted in 1999, main field IGRF models were computed primarily from ground-based observatory data. To determine where the largest errors between satellite field models and models derived from ground-based measurements arise, we examined the difference between IGRF-7 model (valid 1995–2000) coefficients as defined for 1995 and the DGRF model for 1995 from Chambodut et al. (2005). The DGRF model was calculated from the backward projection of high quality satellite data from 2000 onwards, though it should be noted that the back-propagation is imperfect. The RMS difference between the IGRF-7 model and the revised DGRF is 104 nT (up to degree 10) and we have plotted the RMS difference per degree between models in Fig. 3 (dashed black line), again up to degree 10. The IGRF-7 model is relatively good at degrees 1, 2 and 4 but poorer at degrees 3 and 5. Whether this pattern is peculiar to the IGRF-7 model or applies in general to field models derived mainly from ground-based data is not known.
We can now use the absolute differences of the Gauss coefficients between the IGRF-7 and DGRF models (dashed black line) as reasonable estimates for the noise pertubations to apply to an assimilated ‘measurement’ within the EnKF. Similarly, the differences in the Gauss coefficients between the field forecast by the SV from the steady flow model and CHAOS-2 (solid black line) can be used as estimates for the variance of the forecast. The absolute differences are relatively similar in magnitude and somewhat complementary—the errors of the IGRF-7 field model are lower at degree 1 and 2 and similar for degree 3 and 4, while the errors of the flow model SV forecast are lower at degrees 5–12. As the IGRF-7 errors only extend to degree 10, the errors for coefficients of degrees 11 and 12 were set to a value of 0.5 nT. (This is the median of the absolute value of the degree 11 and 12 coefficients of IGRF-10).
In the EnKF, random perturbations to the ensemble states can now be controlled using the absolute differences between the coefficients of IGRF-7 and DGRF for 1995 as the expected standard deviations. Each state coefficient (or Gauss coefficient for this study i.e. ) perturbation is thus drawn from a normal distribution N(0, 1) and scaled by the respective error. We explore the effect of using three different uncertainty assumptions for the ‘measured’ field model input to the filter.
4.3 EnKF simulation results
At every twelfth timestep (i.e. annually), a set of noisy Gauss coefficients was assimilated into the system using Eqs. (3) and (4). The Gauss coefficients, initally generated from the CHAOS-2 model, have Gaussian noise added with a standard deviation derived from the RMS difference of the IGRF-7 and DGRF1995 models to simulate the expected uncertainty of a ground-based field model. The results of three different uncertainty assumptions for the assimilated field model were tested. (Note, the EnKF uncertainty models are given roman numerals to avoid confusion with the IGRF-11 candidate models).
We investigated the outcome of using better field models in the assimilation by reducing the noise added to the CHAOS-2 field coefficients. In uncertainty model II, the ‘measured’ field model errors were sampled from a distribution with spread approximately half as large as the field model errors in I (i.e. the standard deviations were divided by 2). The RMS difference to the CHAOS-2 model in 2009.5 is 47 nT. In uncertainty model III the field model errors are approximately a quarter the size of the model errors in I, with RMS difference to CHAOS-2 in 2009.5 of 34 nT. Thus, as the field model improves, the RMS difference becomes smaller, as would be expected. The RMS difference per degree for models II and III are also shown in Fig. 3 (dashed red line and dash-dot red line with diamond markers, respectively). Due to the random nature of the added noise, some degrees are better fit than others. For example, degree 3 is better fit in model I than model II, despite the average uncertainty being smaller for the latter. At higher degrees (n = 11 and 12), the difference of all the models to the CHAOS-2 model is roughly equal.
Reducing the expected error of the forecast model is not really meaningful as we cannot control the goodness of fit of the forecast in this study. Note also that as each simulation is a random realisation of the errors, the results are, of course, indicative of the potential improvements possible.
5. Conclusions
The RMS difference between the IGRF-10 SV prediction for 2009.5 starting from a model of the field for 2004.5 is estimated to be approximately 102 nT. Thus, the average annual RMS misfit between the IGRF-10 model and the ‘true’ magnetic field was about 21 nT. Using a steady flow model generated from SV data prior to 2004.5 to predict the SV for a similar period of time (2004.5–2009.5) resulted in an average RMS difference of approximately 17 nT/yr. This suggests improvement in the SV prediction may be possible. While there is a large variation between proposed IGRF-11 SV candidate models, the coefficients derived from the steady flow model are within the range of most of the candidate models, indicating it is not an unreasonable estimate.
The RMS difference between the DGRF1995 field model (derived from the backward projection of satellite data) and the IGRF-7 model for 1995 (derived primarily from ground-based observatories) is 104 nT. In a future scenario where only a relatively poor field model is available, we show that it is possible to improve the prediction using data assimilation with a field model derived from a steady core flow forecast. If a high-quality starting field model is available with only lower-quality updates, then the combination of forecasts from a core flow model and lower quality field models using an Ensemble Kalman filter can reduce the RMS errors of the resulting forecast. The potential improvements are strongly dependent on the relative uncertainties of each model.
Declarations
Acknowledgments
We would like to thank Susan Macmillan for her suggestions on using earlier generations of IGRF models for error analysis. We thank the two anonymous reviewers for their helpful comments on the paper. The CHAMP data used in this study were supplied by GFZ Potsdam. This research is part of the NERC GEOSPACE programme, funded under grant NER/O/S/2003/00674. This paper is published with the permission of the Executive Director of the British Geological Survey (NERC).
Authors’ Affiliations
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